AME COMPUTATIONAL MULTIBODY DYNAMICS. Friction and Contact-Impact
|
|
- Preston Whitehead
- 5 years ago
- Views:
Transcription
1 1 Friction AME COMPUTATIONAL MULTIBODY DYNAMICS Friction and Contact-Impact Friction force may be imposed between contacting bodies to oppose their relative motion. Friction force can be a function of the material of the contacting bodies, the shape and the roughness of the surfaces, relative velocity, normal force, lubrication, and other factors. Therefore, determining the force caused by all possible form of friction in one formula is not possible. In general we know friction either as Coulomb or viscous, or a combination of the two. Consider the two bodies shown in Figure 1 being in contact. Let us assume that each body applies a normal force on the other body causing a pair of friction forces,, opposing the relative motion. Coulomb friction, also known as the dry friction, may exist even if the two surfaces do not move relative to each other. If the relative motion between the two bodies is zero, the Coulomb friction is computed as (sf ) f = (1) where is the coefficient of static friction. When the relative motion is nonzero, the friction force is determined as (df ) f = (2) where is the coefficient of dynamic (or kinetic) friction (normally < ). Assuming that the dynamic friction is not a function of the relative velocity, and that the transition from static to dynamic friction is instantaneous, Eqs. (1) and (2) can be presented graphically as shown in Figure 2(a). v i v j (a) (b) Figure 1: (a) Two bodies in contact and (b) the resultant normal and reaction forces. Viscous friction, also known as the wet or lubricated friction, is proportional to the relative velocity of the contacting surfaces. This type of friction can be described in its simplest form as (vf ) f = μ v (3) where μ v is the coefficient of viscous friction, and is the magnitude of the relative velocity = v i v j, or relative speed. If the coefficient of viscous friction is not a constant and is a function of the normal force, then the viscous friction force can be determined as (vf ) f = k v (4) where k v is a different definition of the coefficient of viscous friction. If the friction is assumed to be a combination of Coulomb and viscous frictions, the combined models can be presented as shown in Figure 2(b), where as the relative velocity increases so does the friction force. 1
2 2 AME553 MULTIBODY DYNAMICS v i. j v i. j (a) (b) v i. j v i. j (c) (d) Figure 2: (a) Coulomb friction with static and dynamic coefficients; (b) addition of viscous friction; (c) Stribeck effect; and (d) Approximate representation of static friction. It has been found experimentally that the transition of friction force from zero to nonzero relative velocity is not instantaneous, but it takes place during a short period of time. This transition, known as the Stribeck effect, is depicted in Figure 2(c). Implementation of Coulomb and viscous frictions, including the Stribeck effect, in the equations of motion of a multibody system is not difficult as long as the relative velocity of the contacting bodies is not zero. The difficulty arises when the relative velocity becomes zero, or during the so-called stiction, the static friction force that should be (sf ) f cannot easily be determined in a dynamic analysis. To remedy this problem, the static friction force can be assumed to vary from to during a short range of varied relative velocity in the vicinity of stiction, as shown in Figure 2(d). A variety of analytical formulas to closely represent the force-velocity characteristics of the function of Figure 2(d) can be found in literature. One example of such functions is the following [1]: v p i, j = f v s N + ( )e tanh(k t ) + (vf ) f (5) The general shape of this function is shown in Figure 3. The viscous friction in this formula can be defined as either Eq. (3) or Eq. (4). In addition to the coefficients of friction and the relative speed, Eq. (5) also contains other parameters. Each of these parameters can change the shape of the function. The coefficient of sliding speed, v s, changes the shape of the decay in the Stribeck region as shown in Figure 4(a), where a smaller value for this parameter results in a sharper drop from the static to dynamic friction. The exponent p affects the drop from static to dynamic friction in a different form as shown in Figure 4(b). The parameter k t adjusts the slope of the curve from zero relative speed the maximum static friction, as depicted in Figure 4(c).
3 Friction and Contact 3 Friction force Relative speed Figure 3: Friction force versus relative speed as described by Eq. (5). v s = v s = v s = p = 3 p = 2 p = 1 k t = 10,000 k t = 6,000 k t = 3, (a) (b) (c) Figure 4: Effects of three parameters on the friction force curve. Typical values for the three parameters are v s = m/sec, p = 2, and k t = 10,000. A large value for k t makes the slope of the curve in the static friction region almost vertical. We should also note that with this friction model the maximum value of the static friction force does not reach the desired value. Therefore we may assign a larger value for to compensate for this shortcoming of the model. Another example of an analytical formula that closely represents the force-velocity characteristics of the function of Figure 3 is the following [2]: ( f 16v ) f = tanh(4v r ) + ( ) r ) (6) v 2 r + 3 f nt ( ) 2 + μ v tanh(4 where v r =, and v v t is the transition velocity indicating the velocity at which the friction force reaches t its maximum value and the transition from static to dynamic region begins. The parameter f nt, which is called the transition force, specifies the required minimum normal force for viscous friction to take place. Typical representations of this model for different values of the parameter v t are shown in Figure 5. We note that the transition from static to dynamic friction occurs at the specified transition velocity, at which the friction force is has reached the limit of the static force.
4 4 AME553 MULTIBODY DYNAMICS Friction force v t = v t = v t = Figure 5: Typical friction force-speed curve for the model of Eq. (6). If the normal force is the result of a contact represented as a penetration model, the friction force should be categorized as an applied load. However, if the contact is modeled as a constraint, for which a Lagrange multiplier represents the resulting contact force, the corresponding friction must be categorized as a reaction force. In either case, the contact and friction models can be considered between bodies of different shapes in the contact area, as shown in Figure 6 for a few examples. v i v j Figure 6: Examples of contact with or without friction between different objects. 2 Contact Force (Continuous Analysis) In the continuous analysis method, it is assumed that when two bodies collide, although the contact period is very small, the change in the velocities is not discontinuous the velocities vary continuously during the period of contact as the contacting bodies undergo local deformations. To represent the local deformations in a simplified manner, it is assumed that the colliding bodies penetrate into each other in the contact region as depicted in the examples of Figure 7(a). The amount of penetration, which is denoted as δ, represents the sum of local deformations of the two contacting bodies. The penetration results in a pair of resistive contact forces acting on the two bodies in opposite directions as shown in the illustrations of Figure 7(b). The penetration starts in a compression phase with δ = 0 and reaches a maximum value = m. During the restitution phase, the penetration returns from = m back to δ = 0. Following an impact, the two bodies may remain in contact or they may separate. A contact force model can be viewed as a logical point-to-point spring-damper that is only active during the period of contact. The characteristics of the logical spring-damper element must be adjusted
5 Friction and Contact 5 based on the material properties of the contacting bodies. In the following subsections, we consider several models for a body contacting a rigid surface, and for two bodies contacting each other. δ δ (a) (b) Figure 7 (a) Penetration models for contact-impact, and (b) the corresponding FBD s. 2.1 A Body Contacting A Rigid Surface The simplest contact model to consider is the mass-spring system shown in Figure 8. The single body with the mass m moves toward a rigid surface. As the body contacts the surface it deforms (or penetrates the surface) by an amount. Assuming the deformation is represented by a spring of stiffness k, the penetration force can be determined as = k (7) If we assume that the deformation also contains damping, the penetration force can be computed as = k + d c (8) where d c is the damping coefficient. m k Figure 8 A mass-spring contact model. In the models of Eqs. (7) and (8) it is assumed that both the spring and the damper have linear characteristics. We may also consider nonlinear characteristics for the spring as = k n (9) where n > 1 is an exponent to be determined based on the material characteristics of the body. With such a nonlinear model for the spring, Eq. (8) becomes: = k n + d c (10) In this nonlinear model, a reasonable value for the exponent is n = 3/2 as it will be discussed in the next subsection.
6 6 AME553 MULTIBODY DYNAMICS In the model of Eq. (7) the penetration force vary linearly during both compression and restitution phases as shown as a solid line in Figure 9(a). The force starts from zero at = 0, reaches a maximum value, and then returns along the same straight line back to zero, where at this point the body separates from the contacting surface. When we add damping to the model, as in Eq. (8), the paths of the compression and restitution phases separate as shown in dashed line in Figure 9(a). This curve shows that as the body contacts the surface, there is a nonzero penetration force due to the velocity of the body at the () start of the compression phase. This velocity causes a penetration force = d c, where () denotes the initial penetration speed at contact. The figure shows that the path during the restitution phase, in the presence of damping, is different from that of the compression phase (called hysteresis). We also observe that near the end of the restitution phase, the penetration force becomes zero, at = s, before the restitution phase is completed. This is the instant at which the body separates from the surface while it continues to undeform. At this point, since = 0, the acceleration of the body is zero and, therefore, the velocity of the body is at a maximum. This means that during the final portion of the restitution phase, the model yields a tensile force, as shown in the figure in dotted line, which should not be applied to the body. The contact force versus penetration for the model of Eq. (9) is shown in solid line in Figure 9(b). Similar to the linear model, in the absence of damping, this model provides the same paths for both compression and restitution phases. When damping is included, as in Eq. (10), the model exhibits hysteresis behavior as shown in dashed line. Similar to the linear model, the penetration force becomes zero at = s, which is the instant at which the body leaves the contacting surface. Contact force d c () compression restitution Contact force compression restitution 0 0 s Penetration (a) s Penetration Figure 9 Compression and restitution phases for (a) the linear and (b) the nonlinear models. (b) 2.2 Two-Body Contact The discussion of the previous subsection can be extended to two bodies becoming in contact. Assume that the two bodies shown in Figure 10 are in the process of contacting each other. The force models of Eqs. (7) through (10) can be implemented between the two bodies to determine the contact force. The overall penetration,, can be considered as the sum of the deformations of the two bodies as = The penetration is determined along the axis of contact, which is normal to the contacting surfaces. The stiffness k, for example in the linear model of Eq. (7), can be determined based on the stiffness of the two springs as k = k 1k 2 k 1 + k 2 (11)
7 Friction and Contact 7 In the presence of damping, as in Eq. (8) and Eq. (10), the two bodies separate at = s before they have completely recovered their deformations, as demonstrated in Figure 9. m 1 k 1 k 2 m Figure 10 Two bodies becoming in contact. The spring model with the nonlinear characteristics of Eq. (9) is known as the Hertz contact force model [3]. The model considers the stiffness to depend on the material property and the local geometry of the contacting bodies. For example, for two colliding spheres with radii R i and R j, the parameter k can be determined as R k = i R j 3 (h i + h j ) R i + R ; h i = 1 ν 2 k : k = i, j (12) j πe k where ν k and E k are the Poisson s ratio and Young s modulus of each sphere, and the exponent in Eq. (9) is determined to be n = 3/2. If one of the contacting surfaces is flat (very large radius of R j = ), Eq. (12) becomes 1 4R 2 k = i 3 (h i + h j ) ; h i = 1 ν 2 k : k = i, j (13) πe k A revised form of damping for Eq. (6) considers the damping coefficient to be a function of indentation as [4] d c = μδ n (14) where μ is called the hysteresis damping factor. Substituting this description of the damping coefficient in Eq. (10) results in the following contact force model: = (k + μ ) n (15) Figure 11 illustrates a comparison between this contact force model (solid line) and the force model of Eq. (10). In the revised model, the contact forces at the start of the compression phase and at the end of restitution phase are zero, which is quite different from the original model of Eq. (10). In the revised model, since the contact force does not reach zero until at the end of the restitution phase, the bodies do not separate until the restitution is fully completed.
8 8 AME553 MULTIBODY DYNAMICS Contact force 0 0 Penetration δ Figure 11 Schematic description of contact force versus penetration for Eqs. (10) (dashed line) and (15) (solid line). Although the contact force model of Eq. (15) has changed the physical characteristics of the contact, it provides a useful feature. It has been shown that the damping factor in Eq. (14) can be expressed as a function of the coefficient of restitution, where such a function cannot be found analytically for the damping coefficient in Eq. (10). The coefficient of restitution is defined as the ratio between the relative penetration speeds at the start of the compression phase, ( ) δ, and at the instant of separation, (+) : (+) e = () (16) One description of the damping factor in term of the coefficient of restitution has been suggested as [5] μ = 3k(1 e2 ) 4 () (17) Substitution of this expression in Eq. (15) provides the following contact force model: = k n 1+ 3(1 e2 ) () 4 (18) A graphical description of this force model is shown in Figure 12. For values of the coefficient of restitution smaller than 1.0, the force-penetration curves exhibit hysteresis behavior. The area inside each hysteresis curve denotes the loss of energy during impact. For an elastic impact, that is e = 1.0, the loss of energy is zero. In other words, the relative speed at the moment of separation is equal to the relative speed at the start of contact. Note that Eq. (18) assumes that during both the compression and restitution phases, δ > 0, where the sign of δ is positive during compression (in the same direction as that of ) and negative during restitution. Contact force e = 0.5 e = 0.75 e = 1.0 Penetration
9 Friction and Contact 9 Figure 12 Contact force of Eq. (18) versus penetration for different values of the coefficient of restitution. Typical values for the stiffness parameter k are (1 10) N/m 1.5. The coefficient of restitution can have a value 0 < e < 1.0. For e = 1.0, Eq. (18) becomes identical to Eq. (9); that is, no energy loss due to damping. This represents an elastic impact causing the separation speed to be equal to the approach speed. In contrast e = 0, representing an inelastic (or plastic) impact, results in the maximum loss of energy due to damping. The model of Eq. (18) provides a better representation of contact for values of e closer to 1.0 than the values closer to 0. This shortcoming of Eq. (18) has been improved upon in the following force model [6, 7]: = K n 1+ 8(1 e) 5e (19) In this model, the damping factor becomes larger, compared to that of Eq. (18), as the coefficient of restitution becomes smaller. We note that for e = 0, the damping factor becomes infinity and, therefore, for computational purposes the coefficient of restitution should not be set exactly to zero but it could be a very small number. References 1. Andersson, S., Söderberg, A., and Björklund, S., Friction Models for Sliding Dry, Boundary and Mixed Lubricated Contacts, Tribol. Int., 40(4) (2007) 2. Brown, P., and McPhee, J., A Continuous Velocity-Based Friction Model for Dynamics and Control With Physically Meaningful Parameters, ASME J. Computational and Nonlinear Dynamics, 11(5) (2016) 3. Hertz, H., Gesammelte Werke, Vol. 1, Leipzig (1893) 4. Hunt, K.H., and Grossley, F.R.E., Coefficient of Restitution Interpreted as Damping in Vibroimpact, ASME J. Applied Mechanics, 42(2), (1975) 5. Lankarani, H.M., and Nikravesh, P.E., A Contact Force Model With Hysteresis Damping for Impact Analysis of Multibody Systems, ASME J. Mechanical Design, 112(3), (1990) 6. Ye, K., Li, L., and Zhu, H., A Note on the Hertz Contact Model With Nonlinear Damping for Pounding Simulation. Earthquake Eng. Struct. Dyn. 38, (2009) 7. Flores, P., Machado, M., and Silva, M.T., On the Continuous Contact Force Models for Soft Materials in Multibody Dynamics, J. Mutibody Systems Dynamics, 25: (2011) ()
The Effect of Coefficient of Restitution for Two Collision Models
The Effect of Coefficient of Restitution for Two Collision Models ALACI STELIAN 1, FILOTE CONSTANTIN 2, CIORNEI FLORINA-CARMEN 1 1 Mechanics and Technologies Department 2 The Computers, Electronics and
More informationIdentification of Compliant Contact Force Parameters in Multibody Systems Based on the Neural Network Approach Related to Municipal Property Damages
American Journal of Neural Networks and Applications 2017; 3(5): 49-55 http://www.sciencepublishinggroup.com/j/ajnna doi: 10.11648/j.ajnna.20170305.11 ISSN: 2469-7400 (Print); ISSN: 2469-7419 (Online)
More informationforce model, based on the elastic Hertz theory, together with a dissipative term associated with the internal damping, is utilized to
FLORES, P., LANKARANI, H.M., DYNAMIC RESPONSE OF MULTIBODY SYSTEMS WITH MULTIPLE CLEARANCE JOINTS. ASME JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS, VOL. 7(3), 313-13, 1. ABSTRACT A general methodology
More informationThe... of a particle is defined as its change in position in some time interval.
Distance is the. of a path followed by a particle. Distance is a quantity. The... of a particle is defined as its change in position in some time interval. Displacement is a.. quantity. The... of a particle
More informationSupplementary Material
Mangili et al. Supplementary Material 2 A. Evaluation of substrate Young modulus from AFM measurements 3 4 5 6 7 8 Using the experimental correlations between force and deformation from AFM measurements,
More informationInfluence of the contact impact force model on the dynamic response of multi-body systems
21 Influence of the contact impact force model on the dynamic response of multi-body systems P Flores 1, J Ambrósio 2, J C P Claro 1, and H M Lankarani 3 1 Departamento de Engenharia Mecânica, Universidade
More informationA critical overview of internal and external cylinder contact force models
A critical overview of internal and external cylinder contact force models Cândida M. Pereira, Amílcar L. Ramalho, Jorge A. Ambrósio To cite this version: Cândida M. Pereira, Amílcar L. Ramalho, Jorge
More informationTHE ANALYSIS OF AN ANALOGOUS HUYGENS PENDULUM CONNECTED WITH I.T.C. USING FLEXIBLE MULTIBODY DYNAMIC SIMULATIONS
6 th International Conference Computational Mechanics and Virtual Engineering COMEC 2015 15-16 October 2015, Braşov, Romania THE ANALYSIS OF AN ANALOGOUS HUYGENS PENDULUM CONNECTED WITH I.T.C. USING FLEXIBLE
More information7. FORCE ANALYSIS. Fundamentals F C
ME 352 ORE NLYSIS 7. ORE NLYSIS his chapter discusses some of the methodologies used to perform force analysis on mechanisms. he chapter begins with a review of some fundamentals of force analysis using
More informationTranslational Mechanical Systems
Translational Mechanical Systems Basic (Idealized) Modeling Elements Interconnection Relationships -Physical Laws Derive Equation of Motion (EOM) - SDOF Energy Transfer Series and Parallel Connections
More informationSimulating Two-Dimensional Stick-Slip Motion of a Rigid Body using a New Friction Model
Proceedings of the 2 nd World Congress on Mechanical, Chemical, and Material Engineering (MCM'16) Budapest, Hungary August 22 23, 2016 Paper No. ICMIE 116 DOI: 10.11159/icmie16.116 Simulating Two-Dimensional
More informationA comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids
Alves, J., Peixinho, N., Silva, M.T., Flores, P., Lankarani, H., A comparative study on the viscoelastic constitutive laws for frictionless contact interfaces in multibody dynamics. Mechanism and Machine
More informationComparative Study of Impact Simulation Models for Linear Elastic Structures in Seismic Pounding
Comparative Study of Impact Simulation Models for Linear Elastic Structures in Seismic Pounding N. U. Mate 1, S. V. Bakre and O. R. Jaiswal 3 1 Research Scholar, Associate Professor, 3 Professor Applied
More informationUNLOADING OF AN ELASTIC-PLASTIC LOADED SPHERICAL CONTACT
2004 AIMETA International Tribology Conference, September 14-17, 2004, Rome, Italy UNLOADING OF AN ELASTIC-PLASTIC LOADED SPHERICAL CONTACT Yuri KLIGERMAN( ), Yuri Kadin( ), Izhak ETSION( ) Faculty of
More informationEFFECT OF STRAIN HARDENING ON ELASTIC-PLASTIC CONTACT BEHAVIOUR OF A SPHERE AGAINST A RIGID FLAT A FINITE ELEMENT STUDY
Proceedings of the International Conference on Mechanical Engineering 2009 (ICME2009) 26-28 December 2009, Dhaka, Bangladesh ICME09- EFFECT OF STRAIN HARDENING ON ELASTIC-PLASTIC CONTACT BEHAVIOUR OF A
More informationModeling Mechanical Systems
Modeling Mechanical Systems Mechanical systems can be either translational or rotational. Although the fundamental relationships for both types are derived from Newton s law, they are different enough
More informationFigure 43. Some common mechanical systems involving contact.
33 Demonstration: experimental surface measurement ADE PhaseShift Whitelight Interferometer Surface measurement Surface characterization - Probability density function - Statistical analyses - Autocorrelation
More informationNON-LINEAR VISCOELASTIC MODEL OF STRUCTURAL POUNDING
3 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 004 Paper No. 308 NON-LINEAR VISCOELASTIC MODEL OF STRUCTURAL POUNDING Robert JANKOWSKI SUMMARY Pounding between structures
More informationStudy of Friction Force Model Parameters in Multibody Dynamics
he 4 th Joint International Conference on Multibody System Dynamics Study of Friction Force Model Parameters in Multibody Dynamics Filipe Marques 1, Paulo Flores 1 and Hamid M. Lankarani 2 May 29 June
More informationAnalysis and Simulation of Mechanical Systems With Multiple Frictional Contacts
University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science June 1991 Analysis and Simulation of Mechanical Systems With Multiple Frictional Contacts
More informationChapter 12. Static Equilibrium and Elasticity
Chapter 12 Static Equilibrium and Elasticity Static Equilibrium Equilibrium implies that the object moves with both constant velocity and constant angular velocity relative to an observer in an inertial
More informationValidation of Volumetric Contact Dynamics Models
Validation of Volumetric Contact Dynamics Models by Michael Boos A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Applied Science in
More informationSoftware Verification
EXAMPLE 6-6 LINK SUNY BUFFALO DAMPER WITH LINEAR VELOCITY EXPONENT PROBLEM DESCRIPTION This example comes from Section 5 of Scheller and Constantinou 1999 ( the SUNY Buffalo report ). It is a two-dimensional,
More informationMathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.
Chapter 3 Mechanical Systems A. Bazoune 3.1 INRODUCION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENS Any mechanical system consists
More informationChapter 13. Simple Harmonic Motion
Chapter 13 Simple Harmonic Motion Hooke s Law F s = - k x F s is the spring force k is the spring constant It is a measure of the stiffness of the spring A large k indicates a stiff spring and a small
More informationAn analysis of elasto-plastic sliding spherical asperity interaction
Wear 262 (2007) 210 219 An analysis of elasto-plastic sliding spherical asperity interaction Robert L. Jackson, Ravi S. Duvvuru, Hasnain Meghani, Manoj Mahajan Department of Mechanical Engineering, Auburn
More informationExperimental Investigation of Fully Plastic Contact of a Sphere Against a Hard Flat
J. Jamari e-mail: j.jamari@ctw.utwente.nl D. J. Schipper University of Twente, Surface Technology and Tribology, Faculty of Engineering Technology, Drienerloolaan 5, Postbus 17, 7500 AE, Enschede, The
More informationContact Modeling of Rough Surfaces. Robert L. Jackson Mechanical Engineering Department Auburn University
Contact Modeling of Rough Surfaces Robert L. Jackson Mechanical Engineering Department Auburn University Background The modeling of surface asperities on the micro-scale is of great interest to those interested
More informationA Level Maths Notes: M2 Equations for Projectiles
A Level Maths Notes: M2 Equations for Projectiles A projectile is a body that falls freely under gravity ie the only force acting on it is gravity. In fact this is never strictly true, since there is always
More informationBIOEN LECTURE 18: VISCOELASTIC MODELS
BIOEN 326 2013 LECTURE 18: VISCOELASTIC MODELS Definition of Viscoelasticity. Until now, we have looked at time-independent behaviors. This assumed that materials were purely elastic in the conditions
More informationEffect of Strain Hardening on Unloading of a Deformable Sphere Loaded against a Rigid Flat A Finite Element Study
Effect of Strain Hardening on Unloading of a Deformable Sphere Loaded against a Rigid Flat A Finite Element Study Biplab Chatterjee, Prasanta Sahoo 1 Department of Mechanical Engineering, Jadavpur University
More informationfriction friction a-b slow fast increases during sliding
µ increases during sliding faster sliding --> stronger fault --> slows sliding leads to stable slip: no earthquakes can start velocity-strengthening friction slow fast µ velocity-strengthening friction
More informationMODELING AND SIMULATION OF HYDRAULIC ACTUATOR WITH VISCOUS FRICTION
MODELING AND SIMULATION OF HYDRAULIC ACTUATOR WITH VISCOUS FRICTION Jitendra Yadav 1, Dr. Geeta Agnihotri 1 Assistant professor, Mechanical Engineering Department, University of petroleum and energy studies,
More informationSimulation in Computer Graphics Elastic Solids. Matthias Teschner
Simulation in Computer Graphics Elastic Solids Matthias Teschner Outline Introduction Elastic forces Miscellaneous Collision handling Visualization University of Freiburg Computer Science Department 2
More informationDynamic responses of space solar arrays considering joint clearance and structural flexibility
Research Article Dynamic responses of space solar arrays considering joint clearance and structural flexibility Advances in Mechanical Engineering 2016, Vol. 8(7) 1 11 Ó The Author(s) 2016 DOI: 10.1177/1687814016657945
More informationBoundary Nonlinear Dynamic Analysis
Boundary Nonlinear Dynamic Analysis Damper type Nonlinear Link Base Isolator type Nonlinear Link Modal Nonlinear Analysis : Equivalent Dynamic Load Damper type Nonlinear Link Visco-Elastic Damper (VED)
More informationVertical bounce of two vertically aligned balls
Vertical bounce of two vertically aligned balls Rod Cross a Department of Physics, University of Sydney, Sydney NSW 2006, Australia Received 26 April 2007; accepted 14 July 2007 When a tennis ball rests
More informationFine adhesive particles A contact model including viscous damping
Fine adhesive particles A contact model including viscous damping CHoPS 2012 - Friedrichshafen 7 th International Conference for Conveying and Handling of Particulate Solids Friedrichshafen, 12 th September
More informationA Comparative Study of Joint Clearance Effects on Dynamic Behavior of Planar Multibody Mechanical Systems
2815 A Comparative Study of Joint Clearance Effects on Dynamic Behavior of Planar Multibody Mechanical Systems Abstract Clearance always exists in actual joint due to many uncertainties such as machining
More informationObjectives: After completion of this module, you should be able to:
Chapter 12 Objectives: After completion of this module, you should be able to: Demonstrate your understanding of elasticity, elastic limit, stress, strain, and ultimate strength. Write and apply formulas
More informationwhere G is called the universal gravitational constant.
UNIT-I BASICS & STATICS OF PARTICLES 1. What are the different laws of mechanics? First law: A body does not change its state of motion unless acted upon by a force or Every object in a state of uniform
More informationStructural Dynamics. Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma).
Structural Dynamics Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma). We will now look at free vibrations. Considering the free
More informationBehavior of Concrete Dam under Seismic Load
Behavior of Concrete Dam under Seismic Load A. Hebbouche Ecole Nationale Supérieure d Hydraulique, Blida-Algérie Departement of civil engineering, Saad dahlab university, Blida, Algeria M. Bensaibi Departement
More informationIdentification of nonlinear phenomena in a stochastically excited beam system with impact
Identification of nonlinear phenomena in a stochastically excited beam system with impact A. de Kraker, N. van de Wouw, H.L.A. van den Bosch, D.H. van Campen Department of Mechanical Engineering, Eindhoven
More informationBeams. Beams are structural members that offer resistance to bending due to applied load
Beams Beams are structural members that offer resistance to bending due to applied load 1 Beams Long prismatic members Non-prismatic sections also possible Each cross-section dimension Length of member
More informationT1 T e c h n i c a l S e c t i o n
1.5 Principles of Noise Reduction A good vibration isolation system is reducing vibration transmission through structures and thus, radiation of these vibration into air, thereby reducing noise. There
More informationUniversity of California at Berkeley Structural Engineering Mechanics & Materials Department of Civil & Environmental Engineering Spring 2012 Student name : Doctoral Preliminary Examination in Dynamics
More informationRaymond A. Serway Chris Vuille. Chapter Thirteen. Vibrations and Waves
Raymond A. Serway Chris Vuille Chapter Thirteen Vibrations and Waves Periodic Motion and Waves Periodic motion is one of the most important kinds of physical behavior Will include a closer look at Hooke
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationSTATICS & DYNAMICS. Engineering Mechanics. Gary L. Gray. Francesco Costanzo. Michael E. Plesha. University of Wisconsin-Madison
Engineering Mechanics STATICS & DYNAMICS SECOND EDITION Francesco Costanzo Department of Engineering Science and Mechanics Penn State University Michael E. Plesha Department of Engineering Physics University
More informationKinetics of Particles
Kinetics of Particles A- Force, Mass, and Acceleration Newton s Second Law of Motion: Kinetics is a branch of dynamics that deals with the relationship between the change in motion of a body and the forces
More informationNonlinear Dynamics and Control of Crank-Slider Mechanism with Multiple Clearance Joints
Journal of Theoretical and Applied Vibration and Acoustics 1 (1) 1-1 (15) Journal of Theoretical and Applied Vibration and Acoustics I S A V journal homepage: http://tava.isav.ir Nonlinear Dynamics and
More information3. Kinetics of Particles
3. Kinetics of Particles 3.1 Force, Mass and Acceleration 3.3 Impulse and Momentum 3.4 Impact 1 3.1 Force, Mass and Acceleration We draw two important conclusions from the results of the experiments. First,
More informationENG1001 Engineering Design 1
ENG1001 Engineering Design 1 Structure & Loads Determine forces that act on structures causing it to deform, bend, and stretch Forces push/pull on objects Structures are loaded by: > Dead loads permanent
More informationRigid Body Kinetics :: Force/Mass/Acc
Rigid Body Kinetics :: Force/Mass/Acc General Equations of Motion G is the mass center of the body Action Dynamic Response 1 Rigid Body Kinetics :: Force/Mass/Acc Fixed Axis Rotation All points in body
More informationInstabilities and Dynamic Rupture in a Frictional Interface
Instabilities and Dynamic Rupture in a Frictional Interface Laurent BAILLET LGIT (Laboratoire de Géophysique Interne et Tectonophysique) Grenoble France laurent.baillet@ujf-grenoble.fr http://www-lgit.obs.ujf-grenoble.fr/users/lbaillet/
More informationPHY 101. Work and Kinetic Energy 7.1 Work Done by a Constant Force
PHY 101 DR M. A. ELERUJA KINETIC ENERGY AND WORK POTENTIAL ENERGY AND CONSERVATION OF ENERGY CENTRE OF MASS AND LINEAR MOMENTUM Work is done by a force acting on an object when the point of application
More informationLecture 5. Rheology. Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm
Lecture 5 Rheology Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm WW Norton; unless noted otherwise Rheology is... the study of deformation and flow of
More informationKinetics of Particles: Work and Energy
Kinetics of Particles: Work and Energy Total work done is given by: Modifying this eqn to account for the potential energy terms: U 1-2 + (-ΔV g ) + (-ΔV e ) = ΔT T U 1-2 is work of all external forces
More informationBack Calculation of Rock Mass Modulus using Finite Element Code (COMSOL)
Back Calculation of Rock Mass Modulus using Finite Element Code (COMSOL) Amirreza Ghasemi 1. Introduction Deformability is recognized as one of the most important parameters governing the behavior of rock
More informationStudy on Impact Dynamics of Development for Solar Panel with Cylindrical Clearance Joint
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering00 (2014) 000 000 www.elsevier.com/locate/procedia APISAT2014, 2014 Asia-Pacific International Symposium on Aerospace Technology,
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Essential physics for game developers Introduction The primary issues Let s move virtual objects Kinematics: description
More informationMIT Blossoms lesson on Elasticity: studying how Solids change shape and size Handouts for students
MIT Blossoms lesson on Elasticity: studying how Solids change shape and size Handouts for students Sourish Chakravarty Postdoctoral Associate The Picower Institute for Learning and Memory Massachusetts
More informationA Mass-Spring-Damper Model of a Bouncing Ball*
Int. J. Engng Ed. Vol., No., pp. 393±41, 6 949-149X/91 $3.+. Printed in Great Britain. # 6 TEMPUS Publications. A Mass-Spring-Damper Model of a Bouncing Ball* MARK NAGURKA and SHUGUANG HUANG Department
More information2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity
2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics
More informationDynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras
Dynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras Module - 1 Lecture - 10 Methods of Writing Equation of Motion (Refer
More informationChapter 5. Vibration Analysis. Workbench - Mechanical Introduction ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.
Workbench - Mechanical Introduction 12.0 Chapter 5 Vibration Analysis 5-1 Chapter Overview In this chapter, performing free vibration analyses in Simulation will be covered. In Simulation, performing a
More informationA FINITE ELEMENT STUDY OF ELASTIC-PLASTIC HEMISPHERICAL CONTACT BEHAVIOR AGAINST A RIGID FLAT UNDER VARYING MODULUS OF ELASTICITY AND SPHERE RADIUS
Proceedings of the International Conference on Mechanical Engineering 2009 (ICME2009) 26-28 December 2009, Dhaka, Bangladesh ICME09- A FINITE ELEMENT STUDY OF ELASTIC-PLASTIC HEMISPHERICAL CONTACT BEHAVIOR
More informationThe effect of plasticity in crumpling of thin sheets: Supplementary Information
The effect of plasticity in crumpling of thin sheets: Supplementary Information T. Tallinen, J. A. Åström and J. Timonen Video S1. The video shows crumpling of an elastic sheet with a width to thickness
More informationForce, Mass, and Acceleration
Introduction Force, Mass, and Acceleration At this point you append you knowledge of the geometry of motion (kinematics) to cover the forces and moments associated with any motion (kinetics). The relations
More informationME 274 Spring 2017 Examination No. 2 PROBLEM No. 2 (20 pts.) Given:
PROBLEM No. 2 (20 pts.) Given: Blocks A and B (having masses of 2m and m, respectively) are connected by an inextensible cable, with the cable being pulled over a small pulley of negligible mass. Block
More informationCOMPARISON BETWEEN 2D AND 3D ANALYSES OF SEISMIC STABILITY OF DETACHED BLOCKS IN AN ARCH DAM
COMPARISON BETWEEN 2D AND 3D ANALYSES OF SEISMIC STABILITY OF DETACHED BLOCKS IN AN ARCH DAM Sujan MALLA 1 ABSTRACT The seismic safety of the 147 m high Gigerwald arch dam in Switzerland was assessed for
More informationPLANAR KINETIC EQUATIONS OF MOTION (Section 17.2)
PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) We will limit our study of planar kinetics to rigid bodies that are symmetric with respect to a fixed reference plane. As discussed in Chapter 16, when
More informationWhat make cloth hard to simulate?
Cloth Simulation What make cloth hard to simulate? Due to the thin and flexible nature of cloth, it produces detailed folds and wrinkles, which in turn can lead to complicated selfcollisions. Cloth is
More informationSome Aspects of Structural Dynamics
Appendix B Some Aspects of Structural Dynamics This Appendix deals with some aspects of the dynamic behavior of SDOF and MDOF. It starts with the formulation of the equation of motion of SDOF systems.
More information27. Impact Mechanics of Manipulation
27. Impact Mechanics of Manipulation Matt Mason matt.mason@cs.cmu.edu http://www.cs.cmu.edu/~mason Carnegie Mellon Lecture 27. Mechanics of Manipulation p.1 Lecture 27. Impact Chapter 1 Manipulation 1
More informationDynamics and control of mechanical systems
Dynamics and control of mechanical systems Date Day 1 (03/05) - 05/05 Day 2 (07/05) Day 3 (09/05) Day 4 (11/05) Day 5 (14/05) Day 6 (16/05) Content Review of the basics of mechanics. Kinematics of rigid
More informationMITOCW MITRES2_002S10nonlinear_lec15_300k-mp4
MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources
More informationMechanical Properties of Materials
Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of
More informationSIMULATION OF NONLINEAR VISCO-ELASTICITY
SIMULATION OF NONLINEAR VISCO-ELASTICITY Kazuyoshi Miyamoto*, Hiroshi Yoshinaga*, Masaki Shiraishi*, Masahiko Ueda* *Sumitomo Rubber Industries,LTD. 1-1,2,Tsutsui-cho,Chuo-ku,Kobe 651-0071,Japan Key words;
More informationKinematics 1D Kinematics 2D Dynamics Work and Energy
Kinematics 1D Kinematics 2D Dynamics Work and Energy Kinematics 1 Dimension Kinematics 1 Dimension All about motion problems Frame of Reference orientation of an object s motion Used to anchor coordinate
More informationChapter 7. Highlights:
Chapter 7 Highlights: 1. Understand the basic concepts of engineering stress and strain, yield strength, tensile strength, Young's(elastic) modulus, ductility, toughness, resilience, true stress and true
More informationUNIVERSITY OF MALTA JUNIOR COLLEGE JUNE SUBJECT: ADVANCED APPLIED MATHEMATICS AAM J12 DATE: June 2012 TIME: 9.00 to 12.00
UNIVERSITY OF MALTA JUNIOR COLLEGE JUNE 2012 SUBJECT: ADVANCED APPLIED MATHEMATICS AAM J12 DATE: June 2012 TIME: 9.00 to 12.00 Attempt any 7 questions. Directions to candidates The marks carried by each
More informationLECTURE 9 FRICTION & SPRINGS. Instructor: Kazumi Tolich
LECTURE 9 FRICTION & SPRINGS Instructor: Kazumi Tolich Lecture 9 2 Reading chapter 6-1 to 6-2 Friction n Static friction n Kinetic friction Springs Static friction 3 Static friction is the frictional force
More informationNonlinear rocking of rigid blocks on flexible foundation: analysis and experiments
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 199 (017) 84 89 X International Conference on Structural Dynamics, EURODYN 017 Nonlinear rocking of rigid blocks on flexible
More informationCable-Pulley Interaction with Dynamic Wrap Angle Using the Absolute Nodal Coordinate Formulation
Proceedings of the 4 th International Conference of Control, Dynamic Systems, and Robotics (CDSR'17) Toronto, Canada August 21 23, 2017 Paper No. 133 DOI: 10.11159/cdsr17.133 Cable-Pulley Interaction with
More informationDESIGN AND APPLICATION
III. 3.1 INTRODUCTION. From the foregoing sections on contact theory and material properties we can make a list of what properties an ideal contact material would possess. (1) High electrical conductivity
More informationSteady Flow and its Instability of Gravitational Granular Flow
Steady Flow and its Instability of Gravitational Granular Flow Namiko Mitarai Department of Chemistry and Physics of Condensed Matter, Graduate School of Science, Kyushu University, Japan. A thesis submitted
More informationTHEORY OF VIBRATION ISOLATION
CHAPTER 30 THEORY OF VIBRATION ISOLATION Charles E. Crede Jerome E. Ruzicka INTRODUCTION Vibration isolation concerns means to bring about a reduction in a vibratory effect. A vibration isolator in its
More informationl1, l2, l3, ln l1 + l2 + l3 + ln
Work done by a constant force: Consider an object undergoes a displacement S along a straight line while acted on a force F that makes an angle θ with S as shown The work done W by the agent is the product
More informationNumerical Modelling of Dynamic Earth Force Transmission to Underground Structures
Numerical Modelling of Dynamic Earth Force Transmission to Underground Structures N. Kodama Waseda Institute for Advanced Study, Waseda University, Japan K. Komiya Chiba Institute of Technology, Japan
More informationIdentification of model parameters from elastic/elasto-plastic spherical indentation
Thomas Niederkofler a, Andreas Jäger a, Roman Lackner b a Institute for Mechanics of Materials and Structures (IMWS), Department of Civil Engineering, Vienna University of Technology, Vienna, Austria b
More informationME 2570 MECHANICS OF MATERIALS
ME 2570 MECHANICS OF MATERIALS Chapter III. Mechanical Properties of Materials 1 Tension and Compression Test The strength of a material depends on its ability to sustain a load without undue deformation
More informationLecture 19. Measurement of Solid-Mechanical Quantities (Chapter 8) Measuring Strain Measuring Displacement Measuring Linear Velocity
MECH 373 Instrumentation and Measurements Lecture 19 Measurement of Solid-Mechanical Quantities (Chapter 8) Measuring Strain Measuring Displacement Measuring Linear Velocity Measuring Accepleration and
More informationChapter 4 Analysis of a cantilever
Chapter 4 Analysis of a cantilever Before a complex structure is studied performing a seismic analysis, the behaviour of simpler ones should be fully understood. To achieve this knowledge we will start
More informationLecture 6 mechanical system modeling equivalent mass gears
M2794.25 Mechanical System Analysis 기계시스템해석 lecture 6,7,8 Dongjun Lee ( 이동준 ) Department of Mechanical & Aerospace Engineering Seoul National University Dongjun Lee Lecture 6 mechanical system modeling
More informationA Finite Element Study of the Residual Stress and Deformation in Hemispherical Contacts
obert Jackson 1 Mem. ASME e-mail: robert.jackson@eng.auburn.edu Itti Chusoipin Itzhak Green Fellow, ASME George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA
More informationPROOF COPY [TRIB ] JTQ
AQ: #1 Saad Mukras Assistant Professor Department of Mechanical Engineering, 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Qassim University, Buraydah, Qassim, Saudi Arabia e-mail: mukras@qec.edu.sa
More informationLab Exercise #5: Tension and Bending with Strain Gages
Lab Exercise #5: Tension and Bending with Strain Gages Pre-lab assignment: Yes No Goals: 1. To evaluate tension and bending stress models and Hooke s Law. a. σ = Mc/I and σ = P/A 2. To determine material
More informationChapter 11: Elasticity and Periodic Motion
Chapter 11 Lecture Chapter 11: Elasticity and Periodic Motion Goals for Chapter 11 To study stress, strain, and elastic deformation. To define elasticity and plasticity. To follow periodic motion to a
More information